Seminars and Colloquia by Series

Measuring combinatorial complexity via regularity lemmas

Series
Time
Friday, April 5, 2024 - 16:00 for 1 hour (actually 50 minutes)
Location
Lecture Auditorium 1443, Klaus Building
Speaker
Caroline Terry – Ohio State University

Atlanta Combinatorics Colloquium Hosted by Georgia Tech

Abstract: Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemerédi's regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon–Fischer–Newman, Lovász–Szegedy, and Malliaris–Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies have deep connections to model theory. In this talk, I present extensions of this type of result to arithmetic regularity lemmas, which are analogues of graph regularity lemmas, tailored to the study of combinatorial problems in finite groups. This work uncovered tight connections between tools from additive combinatorics, and ideas from the model theoretic study of infinite groups.

Riemannian geometry and contact topology III

Series
Geometry Topology Working Seminar
Time
Friday, April 5, 2024 - 14:00 for 2 hours
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech

This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series). 

Geometry, topology, and combinatorics of fine curve graph variants

Series
Dissertation Defense
Time
Friday, April 5, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Roberta Shapiro – Georgia Tech

The goal of this talk is to explore curve graphs, which are combinatorial tools that encode topological information about surfaces. We focus on variants of the fine curve graph of a surface. The fine curve graph has its vertices essential simple closed curves on the surface and its edges connect pairs of curves that are disjoint. We will mention a sampling of related theorems which were proven in collaboration with various coauthors and then prove several results regarding the finitary curve graph, which has as its vertices essential simple closed curves while its edges connect pairs of curves that intersect at finitely many points.

In this talk, we will prove that the finitary curve graph has diameter 2 (geometry), that the flag complex induced by the finitary curve graph is contractible (topology), and that the automorphism group of the finitary curve graph is naturally isomorphic to the homeomorphism group of the surface (combinatorics).

Work mentioned in the talk will be a subset of independent work and of collaborations with Katherine Booth, Ryan Dickmann, Dan Minahan, and Alex Nolte. The talk will be aimed at a non-expert audience.

Local vs Non-Local Poincar\'e Inequalities and Quantitative Exponential Concentration

Series
Stochastics Seminar
Time
Thursday, April 4, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian Houdré – Georgia Institute of Technology

Weighted Poincar\'e inequalities known for various laws such as the exponential or Cauchy ones are shown to follow from the "usual"  Poincar\'e inequality involving the non-local gradient.  A key ingredient in showing so is a covariance representation and Hardy's inequality.  

The framework under study is quite general and comprises infinitely divisible laws as well as some log-concave ones.  This same covariance representation is then used to obtain quantitative concentration inequalities of exponential type, recovering in particular the Gaussian results.  

Joint Work with Benjamin Arras.  

On the Curved Trilinear Hilbert Transform

Series
Analysis Seminar
Time
Wednesday, April 3, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bingyang Hu – Auburn University –

The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. We show that it is bounded in the Banach range.  

The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:

1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;

2). a structural analysis of suitable maximal "joint Fourier coefficients";

3). a level set analysis with respect to the time-frequency correlation set. 

This is a joint work with my postdoc advisor Victor Lie from Purdue.

Structure of Boundaries of 3-Dimensional Convex Divisible Domains

Series
Geometry Topology Student Seminar
Time
Wednesday, April 3, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Nolte – Georgia Tech

I read Benoist's paper Convexes Divisibles IV (2006, Invent. Math.), and will talk about it. The main result is a striking structural theorem for triangles in the boundaries of 3-dimensional properly convex divisible domains O that are not strictly convex (which exist). These bound "flats" in O. Benoist shows that every Z^2 subgroup of the group G preserving O preserves a unique such triangle. Conversely, all such triangles are disjoint and any such triangle descends to either a torus or Klein bottle in the quotient M = O/G (and so must have many symmetries!). Furthermore, this "geometrizes" the JSJ decomposition of M, in the sense that cutting along these tori and Klein bottles gives an atoroidal decomposition of M.

Advancements in Persistence Solutions for Functional Perturbed Uniformly Hyperbolic Trajectories: Insights into Relativistic Charged Particle Motion

Series
Math Physics Seminar
Time
Wednesday, April 3, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Joan Gimeno – Universitat de Barcelona –

Please Note: Available online at: https://gatech.zoom.us/j/98258240051

We develop a method to construct solutions of some (retarded or advanced) equations. A prime example could be the motion of point charges interacting via the fully relativistic Lienard-Wiechert potentials (as suggested by J.A. Wheeler and R.P. Feynman in the 1940's). These are retarded equations, but the delay depends implicitly on the trajectory. We assume that the equations considered are formally close to an ODE and that the ODE admits hyperbolic solutions (that is, perturbations transversal to trajectory grow exponentially either in the past or in the future) and we show that there are solutions of the functional equation close to the hyperbolic solutions of the ODE. The method of proof does not require to formulate the delayed problem as an evolution for a class of initial data. The main result is formulated in an "a-posteriori" format and allows to show that solutions obtained by non-rigorous approximations are close (in some precise sense) to true solutions. In the electrodynamics (or gravitational) case, this allows to compare the hyperbolic solutions of several post-newtonian approximations or numerical approximations with the solutions of the Lienard-Weichert interaction. This is a joint work with R. de la Llave and J. Yang.

Spectrahedral Geometry of Graph Sparsifiers (Catherine Babecki, Caltech)

Series
Graph Theory Seminar
Time
Tuesday, April 2, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Catherine Babecki – California Institute of Technology –
We propose an approach to graph sparsification based on the idea of preserving the smallest k eigenvalues and eigenvectors of the Graph Laplacian. This is motivated by the fact that small eigenvalues and their associated eigenvectors tend to be more informative of the global structure and geometry of the graph than larger eigenvalues and their eigenvectors. The set of all weighted subgraphs of a graph G that have the same first k eigenvalues (and eigenvectors) as G is the intersection of a polyhedron with a cone of positive semidefinite matrices. We discuss the geometry of these sets and deduce the natural scale of k. Various families of graphs illustrate our construction.

Ribbon disks for the square knot

Series
Geometry Topology Seminar
Time
Monday, April 1, 2024 - 16:30 for 1 hour (actually 50 minutes)
Location
Georgia Tech
Speaker
Alex Zupan – University of Nebraska - Lincoln

A knot K in S^3 is (smoothly) slice if K is the boundary of a properly embedded disk D in B^4, and K is ribbon if this disk can be realized without any local maxima with respect to the radial Morse function on B^4. In dimension three, a knot K with nice topology – that is, a fibered knot – bounds a unique fiber surface up to isotopy. Thus, it is natural to wonder whether this sort of simplicity could extend to the set of ribbon disks for K, arguably the simplest class of surfaces bounded by a knot in B^4. Surprisingly, we demonstrate that the square knot, one of the two non-trivial ribbon knots with the lowest crossing number, bounds infinitely many distinct ribbon disks up to isotopy. This is joint work with Jeffrey Meier.

A Staircase Proof for Contact Non-Squeezing

Series
Geometry Topology Seminar
Time
Monday, April 1, 2024 - 15:00 for 1 hour (actually 50 minutes)
Location
Georgia Tech
Speaker
Lisa Traynor – Bryn Mawr College

Gromov's non-squeezing theorem established symplectic rigidity and is widely regarded as one of the most important theorems in symplectic geometry. In contrast, in the contact setting, a standard ball of any radius can be contact embedded into an arbitrarily small neighborhood of a point. Despite this flexibility, Eliashberg, Kim, and Polterovich discovered instances of contact rigidity for pre-quantized balls in $\mathbb R^{2n} \times S^1$ under a more restrictive notion of contact squeezing. In particular, in 2006 they applied holomorphic techniques to show that for any {\it integer} $R \geq 1$, there does not exist a contact squeezing of the pre-quantized ball of capacity $R$ into itself; this result was reproved by Sandon in 2011 as an application of the contact homology groups she defined using the generating family technique. Around 2016, Chiu applied the theory of microlocal sheaves to obtain the stronger result that squeezing is impossible for all $R \geq 1$. Very recently, Fraser, Sandon, and Zhang, gave an alternate proof of Chiu’s nonsqueezing result by developing an equivariant version of Sandon’s generating family contact homology groups. I will explain another proof of Chiu’s nonsqueezing, one that uses a persistence module viewpoint to extract new obstructions from the contact homology groups as defined by Sandon in 2011. This is joint work in progress with Maia Fraser.

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